Inorganic Materials

7/31/2019 Inorganic Materials

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Inorganic Materials Chemistry

Core Module 7


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Inorganic Materials Chemistry

6 lectures Module 7 2005-6

Synopsis

Lecture 1: Introduction: Methods of synthesis and analysis

Lecture 2: Some structural properties of solids

Lecture 3: Magnetism I: Interactions between solids and magnetic fields

Lecture 4: Magnetism II: Cooperative magnetism

Lecture 5: Dielectric Properties: Interactions between solids and electric fields

Lecture 6: Superconductivity

Bibliography:

S. Elliot The physics and chemistry of solidsH. M. Rosenberg The solid stateA. R. West Solid state chemistry and its applicationsA. F. Wells Structuralinorganic chemistryL. Smart and E. Moore Solid state chemistry. An introductionA. K. Cheetham and P. Day Solid state chemistry. TechniquesM. T. Weller Inorganic materials chemistry

Associated Courses

AKD Transition metals 1st yearKW Solid state structure 1st yearVC Structure and bonding 1st yearRED Metal-ligand metal-metal bonding 2nd year

KW Surface chemistry 2nd yearMB Diffraction 3rd yearKW Solid state chemistry 3rd year


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Learning Objectives: by the end of the course you should be able to

i) describe methods of synthesis of solid state compounds using examples.

ii) understand the difference between synthesis and modification of materials.

iii) understand the limitations of techniques for the analysis of materials.

iv) explain structure types and polymorphism.

v) explain the main reasons for and types of defects in ionic solids.

vi) calculate the proportion of defects in a solid.

vii) explain what is meant by a non-stoichiometric solid and a solid solution.

viii) understand and explain the behaviour of a paramagnet, antiferromagnet, ferromagnet and

ferrimagnet as a function of temperature.

ix) apply the Curie-Weiss equation and calculate the magnetic moment from the magnetic

susceptibility.

x) explain the effect of domain structure of ferro- and ferrimagnets.

xi) explain the superexchange mechanism for rock salt metal oxides.

xii) describe the spinel structure and estimate the net magnetic moment of ferrimagnets.

xiii) describe the perovskite structure and its relation to ferroelectricity using the tolerance

factor.

xiv) describe the phenomenon of piezoelectricity and pyroelecricity.

xv) describe the electric and magnetic properties of superconductors.

xvi) explain the structural characteristics of cuprates, fullerides and borides.

xvii) qualitatively describe BCS theory of superconductivity.

See Structure Visualisation Package on Year 3 page of departmental website for interactive

structures.


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Lecture 1Introduction

What is materials (solid state) chemistry?

Solid state chemistry is concerned with the synthesis, structure, properties and

application of solids including inorganic and organic materials and their composites.At the interface of chemistry, solid state physics, materials science, ceramics,mineralogy and metallurgy.

Why is materials (solid state) chemistry important?

Society and technology are underpinned by the solid state sciences. For example

Computing (data storage, CD lasers, batteries)Construction (concrete, steels)

Transport (catalytic converters, fuel cells, strong lightweight materials)Chemicals (catalysts, sensors)

Medicine (artificial joints, bones, and muscle)Gems (jewellery, cutting tools, lasers)

Synthesis of solid state compounds

Molecular synthesis

Characteristics:1. Solvents

2. Low synthetic temperatures (typically 80 to 250oC)

3. Kinetic control

e.g. Oxidation of alcohols to carboxylic acids

CH3CH2OH CH3CO2H[O]

Thermodynamic control would lead to CO2 and H2O

CH3CH2OH[O]

+ H2OCO2

Solid state synthesis

Characteristics:

1. Commonly no solvent

2. High synthetic temperatures (typically > 300 oC. Can be as high as 3000oC)

3. Thermodynamic control


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Synthetic methods

Solid to solid synthesis (e.g. ceramic metal oxides)

Heating solid precursors. Furnace, Microwaves (dielectric heating and plasmas)

e.g. B aO + T iO 2 B aT i O 31100 oC , 24 h rs

Ba OTi

Why are such high temperatures usually required?

Ion diffusion:

Ions have to diffuse through a solid. Many bonds need to be broken and formed givinghigh activation energy (c.f. molecular chemistry where usually only one bond isbroken or formed).

R

RR

Nu XXNuvs

Bond lengthening not usually isotropic

Ion hops toadjacent site


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Nucleation andsurface area:Ion diffusion requires starting material particles to be incontact (grain boundaries). Smaller particles give greater reaction rates.

If the structure of the product is different from the starting materials nucleation will beslow lowering the reaction rate.

e.g. M g O + A l2 O 3 M g A l2 O 4

Structure:- rock salt corundum spinel

MgO Al2O3

MgO Al2O3MgAl2O4

Mg2+ Al3+

product interfaces

Growth fronts of product interfaces may move at different rates

a) Interface MgO/MgAl2O4: 2 Al3+ - 3 Mg2+ + 4 MgO MgAl2O4

b) Interface MgAl2O4/Al2O3: 3 Mg2+ - 2 Al3+ + 4 Al2O3 3 MgAl2O4Overall reaction : 4 MgO + 4 Al2O3 4 MgAl2O4

Reaction b) gives 3 times as much product as reaction a)

A particle may also have different surfaces that can react at very different rates.

e.g. MgO

Mg O Mg O

O Mg O Mg

Mg O Mg O

O Mg O Mg

Mg

MgMg

Mg

MgMg

MgMg

Mg

Mg

MgMg

O

OO

O

OO

OO

O

O

OO

{100} face {111} face {111} face

There are no general rules that indicate which surface will be the most reactive.


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Liquid to solid synthesis

Melts: Very little difference to reaction/crystallisation of molecular compounds. Veryhigh temperature is usually required. Single crystals are commonly grown from meltsand knowledge of the relevant phase diagram is beneficial.

Solvothermal methods: e.g. Hydrothermal synthesis of zeolites (aluminosilicates)

NaAl(OH)4(aq) + Na2SiO3(aq) + NaOH + template (NR4OH)

GEL

Nax(AlO2)x(SiO2)y.zH2O.NR4OH(crystals)

stir 25 oC

heat 150oC

heat 400oC in O2 (calcination)

Nax(AlO2)x(SiO2)y (crystals)

The zeolites framework is made up of SiO2 tetrahedra with the occasional substitutionof Al for Si. To compensate the charge difference (Si(+4), Al(+3)) sodium cations are

accommodated on the inner surface of the framework.


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Gas to solid synthesis

Decomposition of volatile precursors

This technique is used extensively for the preparation of thin films.

e.g. metal organic chemical vapour deposition (MOCVD) of GaAs (III/V semiconductor)

The gaseous precursors are decomposed onto a substrate. Decomposition can occur byheating or photolytically.

Modification of solids (ion exchange and intercalation/deintercalation)

Note: Both ion exchange and intercalation/deintercalation occur with conservation of

charge.

Ion exchange is where the lattice ions of a structure are exchanged (replaced).

e.g. Zeolite-A.

Na(zeolite-A) + Ca2+(aq) Ca0.5(zeolites-A) + Na+(aq)

Water softening agent (now used in washing powders instead of phosphates)

e.g. Fast-ion conductor (batteries) Na2O.8Al2O3

Na+ Na+ Na+

Na+ Na+ Na+

Al2O3 spinel block

O O O

Na+ Na+ Na+

Na+ Na+ Na+

Al2O3 spinel block

O O O

Al2O3 spinel block

M+ M+ M+

M+ M+ M+

Al2O3 spinel block

O O O

M+ M+ M+

M+ M+ M+

Al2O3 spinel block

O O O

Al2O3 spinel block

Ag(NO3)(l)

300oC

Immerse Na2O.8Al2O3 in molten Ag(NO3). Can exchange M2+

ions for 2 Na+

ions butdue to electrostatic interactions exchange is much slower.

Ga(CH3)3(g) + AsH3(g) GaAs(s) + 3 CH4(g)


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Intercalationis where species are added to a host structure.

e.g. K3C60 (superconductor < 20 K)

3 K + K3C60400oC

C60

The potassium atoms are accommodated in the interstitial sites between C60molecules. An electron is transferred from the potassium atoms to the C60 molecules.

e.g. LixTiS2 (0 < x < 1) (batteries)

STiS

STiS

Ti

S

S

STiS

STiS

Ti

S

S

Li

Li

Li

Lithium cations are accommodated between TiS2 layers. Expansion perpendicular tothe layers occurs. An electron is transferred from the lithium atoms to the TiS2 layers.


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Analysis of a solid

What would we like to know?

These include1. Composition: stoichiometry(elemental analysis) and particle homogeneity

2. Structure: internal bulk, surface, defects3. Morphology and particle size: shape and size distribution of particles

4. Properties (e.g. magnetic, electrical, dielectric, optical, catalytic, hardness,)

Some analytical techniquesDiffraction: Single crystal if lucky but usually powders.X-ray and Neutron diffraction: crystal structure. Neutrons have a magnetic momentthat can interact with magnetic ions giving diffraction allowing the magneticstructure to be determined. Note that the structure obtained is an average structure.

Solid State NMR: Local structure and dynamics. Need NMR active nuclei.

Microscopy:SEM (scanning electron microscopy): Morphology and particle size distribution.Resolution 0.1-10 m.

TEM and STEM ((scanning) transmission electron microscopy): Electron diffractiongives information about the crystal structure of individual particles includingdefects. Resolution 10-100

EDAXS (Elemental dispersion analyser using X-rays): Elemental analysis ofindividual particles.

STM (scanning tunnelling microscopy): surface structure. Atom manipulation. resolution.

Xenon on Ni (110) surface


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Quantum corral particle in a box

Plus many other spectroscopic techniques

Including:

IR spectroscopy: Local structureElectron energy loss spectroscopy (EELS): Electronic and surface structureX-ray fluorescence spectroscopy (XRF): Elemental analysisExtended X-ray absorption fine structure (EXAFS): Local structure

Summary

X-ray

diffraction

Neutron

diffraction

NMR and

ESR

spectroscopy

SEM TEM EDAXS

Elemental

analysis X

Crystal

structure X X X

Local

structure X X X X

DefectsX X

MorphologyX


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Lecture 2

Some structural properties of solids

CrystallinitySolids exhibit a continuum of order from perfect crystals to totally amorphous.long range order = crystalline (e.g. SiO2 as quartz)short range order = amorphous (e.g. SiO2 as glass).

Structure typesStructures of compounds can be divided into classes. Traditionally each class isnamed after an archetypal compound or mineral.

ball and stick and polyhedral representations of ReO3 structure type

e.g.1 ReO3 is built from vertex sharing ReO6 octahedra. Any other compound thatexhibits this type of structure (e.g. NbF3) is called a structure type.We say NbF3 has the ReO3 structure type.e.g.2 The rock salts. These are simple cubic structures that many alkali metal halidesadopt e.g. NaCl. We say NaCl has the rock salt structure type.

Polymorphism

Rutile Anatase Brookite


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Some elements have allotropes e.g. C (graphite, diamond, fullerenes) and somemolecular compounds have isomers e.g. (alkanes).Some solid compounds can also have more than one crystalline structure for a singlestoichiometry. This is called polymorphism and individual structures are calledpolymorphs. Rutile, anatase and brookite are all polymorphs of TiO2.Often a solid having a particular crystalline structure and stoichiometry is referred to

as aphase.

Dimensionality and porosity

Dimensionality is important in understanding the properties of solids.e.g. C60, nanotubes, graphite and diamond.

Carbon nanotube (1-D). SemiconductorsImmense tensile strength.

C60 (0-D). Soluble molecule.

Graphite (2-D). Soft semimetal. Diamond (3-D). Hard insulator.


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Defects and non-stoichiometry in ionic solids

Note: overall charge balance must be maintained.

Only at 0 K will an ionic crystal have a perfectly ordered array of atoms where everyatomic lattice point contains an atom.At > 0 K crystals contain defects. Defect formation requires energy and is always

endothermic.However a structure with defects has higher entropy.Increased entropy drives defect formation to G = 0 (equilibrium).Remember G = H - TS. Strong temperature dependence on the number ofdefects.

0

0 number of defects

energy

TS

H

G

Point defects (intrinsic)

Two types:Schottky defects: vacancies are present in the lattice.Frenkel defects: vacancy is created by an ion (usually cation) moving into an

interstitial site.The lattice will distort at the vacancy site to minimise the loss in lattice energy. Vacancies or interstitial atoms do not have to be adjacent and can be mobile.

Schottky Frenkel

nS = Nexp(-HS/2kT) nF = (NNi)1/2exp(-HF/2kT)

nF Nexp(-HF/2kT)

Where nS (nF) = Shottky (Frenkel) defects per unit volume, HS ((HF) = Shottky(Frenkel) enthalpy of formation, N = number of cation and anion sites per unitvolume, Ni = number of interstitial sites per unit volume.


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Temperature (K) Compound (defect) H (10-19 J) n/N300 MgO (S) 10.57 3.62 x 10-56

300 CaO (S) 9.77 5.69 x 10-52

300 LiF (S) 3.75 2.14 x 10-20

300 LiCl (S) 3.40 1.46 x 10-18

300 LiBr (S) 2.88 7.83 x 10-16

300 LiI (S) 2.08 1.23 x 10-11

300 UO2 (cation F) 5.45 2.59 x 10-29

300 CaF2 (anion F) 4.49 2.81 x 10-24

300 AgCl (cation F) 2.56 3.73 x 10-14

300 AgBr (cation F) 1.92 8.49 x 10-11

300 AgI (cation F) 1.12 1.33 x 10-6

500 LiCl 3.40 1.99 x 10-11

750 LiCl 3.40 7.35 x 10-8

1000 LiCl 3.40 4.46 x 10-6

The percentage of intrinsic point defects in most ionic compounds is small but theycan have a significant effect on electrical, magnetic and optical properties. Thesmallest H(HS or HF)will determine if Shottky or Frenkel defects dominate.

Point defects (extrinsic)

Introducing different ions into the structure can increase the proportion of defects.This is known asdoping.

e.g. In solid state chemistry you will study semi-conductors.Silicon (group 14) is not a good conductor of electricity. If Si is doped with smallamounts (1%) of P (group15) the conductivity increases significantly. In effect afterforming bonds to silicon the remaining electron from P acts as a conduction electron.

Non-stoichiometry and solid solutions

In non-stoichiometric solids intrinsic and extrinsic defects can occur but too a muchlarger extent.e.g. LixTiS2. Non-stoichiometry is common for compounds of transitionmetals that can have variable oxidation state.

e.g. TiO is non-stoichiometric. It has the rock salt structure over the range TiOx(where 0.7 < x < 1.25).Non-stoichiometric compounds are distinct from other compositions e.g. TiO2 becausethey have a common structure.However the lattice parameters (size of the unit cell) of the structure will changegradually.


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Ti O

cellvolume

x1 1.25

Consider that TiO1.25 can be rewritten as Ti0.8O. Is the non-stoichiometry due to extraoxygen present in interstitial sites or vacancies on the titanium site?

From experimental evidenceAt x < 1 oxygen vacancies are present in the structureAt x > 1 titanium vacancies are present in the structure

TiO has both Ti and O vacancies!What is the average oxidation state of Ti in TiO1.25 ?Ti2.5+ = 50:50 mixture of Ti2+ and Ti3+

Solid solutions are commonly observed in non-stoichiometric solids. A solid solution is acrystalline solid that can have continual variable composition for a given structure type.They are a useful method of discovering and fine-tuning properties.

There are two types:substitutional where a new atom replaces an existing atom. e.g. bydoping.

e.g. Al2O3 and Cr2O3 can form a substitutional solid solution over the compositionalrange (Al2-xCrx)2O3 0 < x < 2

600

8001000

1200

1400

1600

1800

2000

2200

solid solution

liquid

two solidsolutions

Al2O3 Cr2O3mole %

T (oC)

Immiscibilitydome

Interstitial. Atoms are added. e.g. LixTiS2 and C to Fe giving FeCx 0 < x < 0.09 (C atomsoccupy the interstitial sites of Fe). (stainless steel).


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Lecture 3

Magnetism: Interactions between solids and magnetic fields

Transition metals and lanthanides can have partially filled valence orbitals that resultin unpaired electrons. The angular momentum (J) of unpaired electrons gives rise tomagnetic behaviour. Magnetic behaviour is a quantum mechanical phenomenon.

We will only consider solids that have localisedelectrons (spins) (i.e. magnetism ofinsulators).We will not consider solids that have itinerant electrons (i.e. the magnetism ofmetals).

Microscopic magnetism

B

Metal Ion

(B = magnetic field experiencedby the ion)

On the microscopic level individual ions with unpaired electrons have a magneticdipole moment . The size of depends on the spin and orbital angular moments.

We will look at how in a solid these individual dipoles can interact with each otherand behave collectively in a magnetic field.

Types of BULK magnetism

Diamagnetism All solids exhibit diamagnetism. Atoms or molecules with closedshells of electrons are diamagnetic. A diamagnet is repelled by an external magneticfield. It is a result of a perturbation of the orbits of all electrons and is not due tomagnetic dipoles. It is a weak phenomenon except in the case of superconductors.Diamagnetism is temperature independent.


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Paramagnetismis observed for compounds that contain unpaired electrons and wherethe magnetic dipoles do not interact with each other. A paramagnet is attracted to anexternal magnetic field.

Due to thermal energy the dipoles are orientated randomly but as the temperaturedecreases there is less thermal energy and the dipoles can begin to align (order)parallel (lowest energy configuration) to an external field.

'low' T'high' T, no H H or very large H

Paramagnetism is temperature dependent.

Antiferromagnetismis an example of cooperative magnetism where no applied fieldis necessary to align dipoles (spins) anti-parallel.

Ferromagnetism is an example of cooperative magnetism where no applied field isnecessary to align spins parallel.

Ferrimagnetism is an example of cooperative magnetism where no applied field is

necessary to align spins. The alignment is essentially non-parallel giving partialcancellation of up and down spins.


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Magnitude and behaviour ofmolfor different types of magnetism

Behaviour typical value ofmol (m

3 mol-1)change ofmolwith increasing

temperature

Field dependence(hysterisis) ?

Diamagnetism -1 x 10-6 none no

Paramagnetism 0 to 10-2

decreases noAntiferromagnetism 0 to 10-2 increases maybeFerromagnetism 10-2 to 106 decreases yesFerrimagnetism 10-2 to 106 variable yes

Some Theory

Some units and definitions

H = external magnetic field intensity (Am-1)M = magnetisation or magnetic dipole moment per unit volume (Am-1)B = internal magnetic field in the sample (NA-1m-1, or Tesla (T), Gauss (G) 1T = 10000G)v = volume magnetic susceptibility (dimensionless)mol = molar magnetic susceptibility (m

3 mol-1, emu mol-1) sometimes called

B = o H

B = o (H + M)

o vacuum permeability = 4 x 10-7 NA-2

v =

(how in a vacuum B and H are related)

(how B and H are related in a material)

the magnetisation is usually presented in terms of the magenticsusceptibility .

therefore B = o H(1 + v)

MH

M = 0-1 v

B

(how the magnetisation (M) is related to the internal magnetic field (B)

or B = o rH (where(1 + v) = r (the relative

permeability)

and


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How does vary with temperature for different types of magnetism?

paramagnet ferromagnetantiferromagnet

T T T

TN TC

TN is the Nel temperature (antiferromagnetic-paramagnetic transition temperature)TC is the Curie temperature (ferromagnetic-paramagnetic transition temperature).

Paramagnetism

The Curie law gives the temperature dependence of for a paramagnet.The Curie-Weiss gives the temperature dependence of for the paramagnetic regionof materials that exhibit a transition between paramagnetic-antiferromagnetic orparamagnetic-ferromagnetic states.

= C = Curie Constant T - = =Weiss constantC

T

C

1

(TN) (TC)

slope C-1

T

In an experiment we measure the temperature dependence of M in an applied field.Plotting -1 vs T gives the effective magnetic moment eff. (unit = (Bohr magneton),1 = 9.274 x 10

-24 JT-1)


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mol = 3kT

NA = Avagadro's constantk = Boltzmann's constant eff

2

B = Bohr magneton

NB2

C =3k

eff2NB

2

Only applicable to cases where eff is temperature independent

eff is related to the number of unpaired electrons:

Spin only formula Includes spin orbit coupling (Russell-Sanders coupling)

S = total spin angular momentum (No. electrons/2), L = total orbital angularmomentum, g = g-factor (free electron value ~2 can be used when L = 0)

ion Unpaired electrons S (calc) S + L (calc) eff(observed)V4+ 1 1.73 3.00 ~ 1.8V3+ 2 2.83 4.47 ~ 2.8Cr3+ 3 3.87 5.20 ~ 3.8Mn2+ 5 (high spin) 5.92 5.92 ~5.9Fe2+ 4 (high spin) 4.90 5.48 5.1 5.5Co2+ 3 (high spin) 3.87 5.20 4.1 5.2Ni2+ 2 2.83 4.47 2.8 4.0Cu2+ 1 1.73 3.00 1.7 2.2

The orbital angular momentum is due to the motion of the electron about the nucleus.In many compounds (particularly the first row) the orbital angular momentum isalmost entirely quenched because the d-orbitals are no longer degenerate.

For heavier transition metals and the lanthanides the spin-orbit coupling is large andRussell-Sanders coupling inappropriate. Use j-j coupling scheme instead. Large spin-orbit coupling can give rise to very large moments. We shall not consider j-j couplingany further.

S = g (S(S +1)) S+L = (4S(S +1) + L(L+1))


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Lecture 4

Cooperative magnetism

We can consider a paramagnetic compound as the magnetic equivalent of an ideal gas. There is no interaction between the particles (magnetic dipoles) and no order.

If we decrease the temperature of a gas the interaction between molecules increases

until order is introduced at the freezing point.Similarly lowering the temperature of solids increases the interactions betweenmagnetic ions increasing the order.The ordering of magnetic ions is cooperative. (transition over small T)

FerromagnetismIs usually associated with metals that have an electronic band structure.You will learnmore about metals and band structures in solid state chemistry in the summer term.

There are only a few elements that are ferromagnetic Fe, Co, Ni, Gd and Tb.

Due to an exchange interaction between dipoles a ferromagnet has all the spinsaligned in parallel in the absence of an external magnetic field.

The alignment can be broken by heating at temperatures > TC. Fe has a TC of 1043K.

T > Tc

If Fe has a TC of 1043K why is not all Fe magnetic at room temperature?and why does placing a magnet next to Fe magnetise it?

Domains

Below TC ferromagnets are divided into domains.

Each domain has ions aligned in parallel, but in the absence of an external magneticfield the domains are not parallel to each other. Entropy drives the formation ofdomains.

On the application of an external magnetic field the domains align in the same

direction as the field.

When all the domains are parallel the magnetisation M is at a maximum (saturationmagnetisation)


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H

M

hard

a

bc

d

H

M

soft

The domains can be aligned in any direction by changing the direction of the externalfield. Aligning and changing the direction of domains requires energy.

a to b. The domains of a sample are aligned in a magnetic field. At b the sample has

reached saturation magnetisation.b to c. The field is returned to 0 but the sample remains magnetised this is known asthe remanent magnetisation.c to d In order to demagnetise the sample a field in the opposite direction is required.This is called the coercive field.

These M vs H diagrams are known as hysterisis curves

A hard magnet has a large remanent magnetisation and large coercive field: used aspermanent magnets and magnetic memory (CrO2).

A soft magnet has a small remanent magentisation and a small coercive field: used astransformer cores.

Antiferromagnetism is observed for many metallic and insulating solids.These include Cr, MnO, FeO, CoO, NiO, FeF2.

At low temperatures most paramagnetic compounds become antiferromagnetic ( =0).The antiparallel alignment can be broken at temperature > TN.

T > TN

In contrast to paramagnets and ferromagnets increasing the temperature of anantiferromagnet increases until TN.


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How do the ions couple antiferromagnetically in metal compounds?

SuperexchangeSuperexchange is where anions mediate magnetic exchange between metal cations.Anions can be sulfur and fluorine but more commonly oxygen.

Superexchange occurs via overlap of atomic orbitals of the oxygen (p) and metal (d)atoms.

e.g. First row transition metal monoxides such as NiO and MnO adopt the rock saltstructure where the metal and oxygen are in octahedral sites.

eg

t2g

Ni NiO

Ni

O

Ni2+

Two eg (dx2-y2 or dz2) orbitals containing an unpaired electron overlap with an O2-p

orbital that contains two oppositely aligned electrons. The spins align themselves to beantiparallel giving an overall antiferromagnetic exchange coupling.

Effectively there are two Ni lattices that contain Ni spins up and one down. Eachof these is know as a sublattice. Below TN the sublattice structure of NiO can bedetermined experimentally using neutron diffraction.

Above TN thermal energy is greater than the superexchange interaction and the spinsno longer align giving paramagnetic behaviour.

FerrimagnetismConsider an antiferromagnet that has two sublattices but where the magnetic momentof one sublattice is greater than the other.

This will result in a net spontaneous magnetisation (M) > 0. As M > 0 the solid is nolonger an antiferromagnet. These compounds are known asferrimagnets.

Ferrimagnets are very important industrial materials because they have similarproperties to ferromagnets. However in contrast to ferromagnets there are manyexamples of electrically insulating ferrimagnets.


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Several examples of ferrimagnets have the spinel and inverse spinel structure.

Spinel is the name given to a structure type of the formula AB2X4 e.g. MgAl2O4.

The structure is based on a cubic close packed array of anion atoms.

The spinels and inverse spinels are then constructed by filling 1/8 of the tetrahedraland of the octahedral sites in an ordered manner.

spinel Inverse spinel

[A2+]tet [B3+]oct [B3+]tet[A2+, B3+]oct

MgAl2O4 CoFe2O4ZnFe2O4 NiFe2O4Mn3O4 Fe3O4 (magnetite) (loadstone)

Not all spinels have clear preference of metal ions for particular coordination sites.Many examples contain mixed metal sites. e.g. MnFe2O4 contains 80% normal and20% inverse.

The preference for normal or inverse spinel is due to several factors.

1. From electrostatics M3+ should prefer the octahedral site and M2+ the tetrahedral site.2. The smaller cation would generally go in the smaller tetrahedral site3. Crystal field stabilisation energies(CFSE)

e.g. NiFe2O4Oxide lattice provides a weak field for the metal atoms.Ni(II) d8 and Fe(III) d5 are high spin. CFSE for Fe in tetrahedral and octahedralcoordination is 0. CFSE for Ni is octahedral > tetrahedral. Therefore Ni goes inoctahedral site leading to the inverse spinel structure.

Tetrahedral sites

Octahedral sites

Oxygen


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Estimation of the magnetic moment of ferrimagnets

Using = gS as a rough estimate (instead ofs = g (S(S +1)) ) we can calculate themagnetic moment for each ion. The saturation (maximum) magnetic moment(sat)for a ferrimagnet is then a vector sum of the individual ion moments.

We need to know how the spins couple with respect to each other.

1) The coupling between octahedral site ions is weak. Between tetrahedral sites alsoweak.2) Superexchange between the octahedral and tetrahedral ions is strong givingantiferromagnetic alignment.Therefore all A moments in same direction and all B moment in same direction (butopposite to A)

Tetrahedral sites

Octahedral sites

e.g. NiFe2O4 (inverse spinel)

ion [Ni2+]oct, d8 [Fe3+]oct, d5 [Fe3+]tet, d5 totNo. unpaired

electrons2 5 5

ion 2 5 5 2 + 5 5 = 2e.g. Mn3O4 (normal spinel)

ion [Mn2+]tet, d5 2 x [Mn3+]oct, d4 totNo. unpaired

electrons5 2 x 4

ion 5 2 x 4 8 5 = 3


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e.g. Fe3O4 (inverse spinel)

ion [Fe2+]oct, d6 [Fe3+]oct, d5 [Fe3+]tet, d5 totNo. unpaired

electrons4 5 5

ion 4 5 5 4 + 5 5 = 4

Comparison of estimation and experimental data forMIIFeIII2O4

M(II) S sat calc sat expt TN Discrepancy

Mn 5/2 5 4.55 573 90% normalFe 2 4 4.1 858 s.o. couplingCo 3/2 3 3.94 793 s.o. coupling

Ni 1 2 2.3 858 s.o. couplingCu 1/2 1 1.3 726 s.o. couplingMg 0 0 1.1 713 90% inverse


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Lecture 5

Dielectric Properties: Interactions between solids and electric fields

One way of classifying solids is their ability to conduct electrons or ions.

material

(ohm-1

cm

-1

)metals 101 105

semiconductors 10-5 102electronic conductioninsulators < 10-12

ionic crystals


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Note: In the absence of an electric field most solids do not have bulknet separation ofcharge and therefore have no overall dipole moment. Crystals of molecules withpermanent dipoles e.g. H2O in ice, tend to have structures where the dipoles cancelout.

The average dipole moment per unit volume is called thepolarisation(P).

P is related to the applied electric field E by the equation:

P = 0eE (c.f. M = o-1v B)

where 0 is the vacuum permittivity (8.8542 x 10-12 J-1 C2 m-1

and eis the dielectric susceptibility.

Commonly the dielectric constant (relative permittivity) r = 1 + e is quoted. (c.f.r)

Most ionic solids have r between 5 and 10.

r is dependent on the temperature and the frequency of the field (if an alternatingelectric field is applied).

On polarisation a material stores charge. The stored charge can be measured in aparallel plate capacitor and r determined.

A good dielectric material should have high dielectric strength (not breakdown at highvoltages and become conducting) and have low dielectric loss (not lose electrical

energy as heat).

dielectric

+ + + + + +

+ + + + + +

- - - - - - - - - - - -

- - - - - - - - - - - -

positive plate

negative plate

r =Cdielectric

Cvacuum

Q = C V

where V = potential differenceQ = stored chargeC = capacitance


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Ferroelectrics, piezoelectrics and pyroelectrics

An applied electric field gives rise to polarisation and a dipole moment. For manymaterials when the electric field is removed the polarisation is lost (paraelectricity).Most materials have structures where ions are in an asymmetric site (and henceasymmetric local electric field) and therefore have a local dipole moment. However

these will cancel in the bulk material unless the structure is non-centrosymmetric.Ferro-, piezo- and pyroelectric materials all have non-centrosymmetric structures.

Ferroelectrics can retain polarisation after the electric field has been removed andhave a very high dielectric constant.

An example of a ferroelectric is BaTiO3 that has the perovskite ABO3 structure(related to ReO3).

r BaTiO3 = 102 104

Ba O Ti

Ti Coordination Ba Coordination

At temperatures above 120oC the Ti atoms are in a symmetric octahedral TiO6 site(cubic BaTiO3 (centrosymmetric)).Between 5 - 120oC the Ti atoms are displaced along one of the axis of the octahedron(by 0.1 ) and polarisation results (tetragonal BaTiO3 (non-centrosymmetric)).

Ba O

Ti

cubic tetragonal

T > 120oC 5 < T < 120oC


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Why does the structure distort?The structure of a compound is dependent on the size, charge and preferredcoordination number/geometry of the constituent ions.For compounds that contain several elements the ideal preferences for each ion maynot be accommodated and strain may be present. Significant strain will result in astructural distortion.

For transition metal oxides distortions can be estimated using a tolerance factor.Consider the perovskite ABO3 structure.

From the diagram it can be seen thata = 2 (rB + rO) and a = 2 (rA + rO)

Therefore to obtain ideal contact between A, Band O

2 (rB + rO) = 2 (rA + rO)

Some deviation from this ideal can be included byintroduction of a tolerance factort.

t 2(rB + rO) = 2 (rA + rO)

For the ideal structure t = 1. As t deviates from 1 strain is introduced due to one orboth of the cations not fitting properly. 0.85 < t < 1.06 gives a distorted perovskite.Outside this range a non-perovskite structure is usually adopted.

e.g. SrTiO3 Sr2+ (r = 158 pm), Ti4+(r = 74.5 pm), O2- (r = 126pm) t = 1.002 (good fit,

almost ideal)

BaTiO3 Ba2+ (r = 175 pm), t = 1.06 (Ti ion is occupying a site larger than it wouldprefer, strain)

At T > 120oC the thermal motion of the Ti atom creates enough chemical pressure toretain the ideal cubic perovskite structure.At 5 < T < 120oC the thermal motion no longer compensates for the strain and thestructure distorts.

The temperature at which a solid becomes ferroelectric is called the Curie temperatureTC. Above TC the solid is paraelectric and the dielectric constant follows Curie-Weissform. c.f. paramagnetic-ferromagnetic transition.

C = Curie ConstantT - r =

=Weiss constantC

rA

+ rO

rB + rO

a


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Do the dipoles point in the same direction? (Comparison between ferroelectricity

and ferromagnetism).

For a ferromagnet, lowering the temperature below TC leads to ordering of the spinsinto domains. Application of an external magnetic field leads to alignment of thedomains.

Lowering the temperature of a ferroelectric below TC leads to a distortion in thestructure.However below TC the dipoles of ferroelectrics do not always form domainsbecause the dipole-dipole interaction is relatively weak.An applied electric field orders dipoles into domains. Application of a strongerexternal electric field then aligns the domains.

Due to the domain structure ferroelectrics exhibit hysterisis..

E(V)a

b

d

P

c

a to b. On application of an electric field individual dipoles are aligned into domainsand subsequently the domains are aligned. At b the sample has reached saturationpolarisation.

b to c. The electric field is returned to 0 but the sample remains polarised this isknown as the remanent polarisation.c to d In order to depolarise the sample a field in the opposite direction is required.This is the coercive field.

It is also possible to have antiferroelectric and ferrielectric behaviour analogous to thatobserved in magnetism.

Ferroelectrics are used mainly as capacitors because near the paraelectric-ferroelectrictransition large quantities of charge can be stored.


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Piezoelectrics polarise under the action of mechanical stress and develop electricalcharges on opposite crystal faces (i.e. a voltage difference). Conversely when anelectric field is placed across a piezoelectric crystal it develops strain.

Many compounds composed of tetrahedral groups distort under stress and arepeizoelectric. Also tetrahedra do not have a centre of symmetry and often lead to non-

centrosymmetric structures. An example is -quartz.

Piezoelectrics are used extensively as transducers for loudspeakers, earphones, inkjetprinters, STM, cigarette lighters etcIf an alternating electric field is applied to a piezoelectric the crystal will vibrate.Resonance occurs if the vibration corresponds to a fundamental mode of the crystal.This oscillation is the basis of using quartz for timekeeping.

Pyroelectrics exhibit a net bulk spontaneous polarisation that is temperaturedependent. Thermal expansion or contraction of the lattice changes the size of thedipoles.

An example of a pyroelectric material is ZnO that has the wurtzite structure.

The ZnO4 tetrahedra (dipoles) point in the same direction, giving rise to a net bulkpolarisation. In contrast to ferroelectrics the polarisation of pyroelectrics cannotusually be reversed by the action of an electric (coercive) field.

Pyroelectrics are used in systems where an electrical response to temperature is useful.e.g. infrared radiation detectors (night vision).

p


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Relationship between ferroelectrics, pyroelectrics, and piezoelectricsPolarisation is responsible for the observed behaviour in all cases.Pyroelectric and piezoelectric are also ferroelectric.Pyroelectric materials are piezolelectric.Not all piezoelectric materials are pyroelectric.


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Lecture 6

Superconductivity

Superconductors exhibit two remarkable properties.1. Below a critical temperature TC zero electrical resistance is observed.

(omh cm-1)

metal

superconductor

T (K)Tc 2. Below TC magnetic flux is expelled B = 0 (Meissner effect). Therefore = -1 (a

superconductor is considered a perfect diamagnet).

above TC below TC

Superconductors are used for the generation of large magnetic fields (such as NMRand MRI) but if the TC could be increased to > 300K many technologies (e.g. transportand power generation) would be revolutionised.Superconductivity was discovered by Onnes in 1911 for Hg (TC = 4.2K) and manyexamples of superconductors have since been investigated. However until 1986 thehighest known TCs were approximately 25K.

Cuprates (high TCsuperconductors)

In 1986 the compound La1.8Ba0.2CuO4 was reported to have a TC of 35K. Thiscompound has the K2NiF4 type structure that is related to the perovskites. Later thephases YBa2Cu3O7-x 0


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Structure of Cuprates

There are two copper sites: A distorted square pyramid of Jahn-teller distorted Cu2+

(that form CuO2 layers) and square planar geometry (that link the layers). Oxygen is

removed from the basal planes (0, , 0) for the compositions YBa2Cu3O7-xleading tolinear geometry for the Cu atoms linking the CuO2 layers. The structure isorthorhombic ab c.The average oxidation state for Cu in YBa2Cu3O7-x 0


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Studies indicate that the important structural features are CuO2 square planar layersseparated by charge reservoir layers that control the average Cu oxidation state inthe CuO2 layers. The average Cu oxidation state should also be > 2+. For YBa2Cu3O7the charge reservoir layers are CuO chains.

Fullerides

Synthesised by intercalation of electropositive metals into the C60 lattice.Electron transfer from the metal to C60 gives C60

n- anions (fullerides). The orbitals ofneighbouring C60 molecules overlap forming bands and the electrons are able to move

throughout the solid i.e metallic. TC high for molecular based solid.

For the superconducting A3C60 (A = alkali metal) that have a cubic structure,all thetetrahedral and octahedral interstitial sites are filled.TC is proportional to the averagecation volume (the distance between C60 molecules).

CuO2 planes

Charge reservoir layer

e-'s

10 20 30 40 500

5

10

15

20

25

30

35

TC (K)

Cation Volume (3)

Cs2Rb

Rb3

Rb2KK2Rb

K3

Na2Cs

Na2Rb0.5Cs0.5


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Borides

In 2002 a new class of superconductor was discovered. MgB2 (TC = 39K) is anintermetallic compound i.e. B atoms are in the intersticies of metallic Mg. The searchis now on to investigate other intermetallic compounds for superconductivity andmodify MgB2. The structure is very simple and is based on hexagonal layers of Mgand B.

Theory of superconductivityThe theory(ies) of superconductivity are mathematically very complicated however aqualitative picture of one (BCS theory) can be given.

BCS (Bardeen, Cooper, and Schrieffer) theory describes the mutual attraction of twoconduction electrons mediated by lattice vibrations.In the same way electromagnetic waves can be quantised as photons, the frequencies

of lattice vibrations can also be quantised asphonons. Electron-phonon coupling is themechanism by which electrons can be attracted to each other. The two electrons areknown collectively as a Cooper pair.

The two electrons do not have to be close to each other and can be many thousands ofatomic spacings apart.

e-

++ + +

++ + +

++ + +

++ + +

e-++ + +

++ + +

++ + +

++ + +

e-

e-

++ + +

++ + +

++ + +

++ + +

Electron-phonon coupling

B

Mg

c b

a


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The importance of phonons can be demonstrated experimentally by the use ofisotopes. TC is inversely proportional to M where M is the mass of a lattice ion.

Except for at very low temperatures the resistance of metals is due to the scattering ofelectrons by phonons, which is why the resistance of metals increases with increasingtemperature. (more phonons at higher temperature). For superconductors Cooper pairs

are not scattered and therefore have zero resistance.

There is of course a repulsive electrostatic interaction between two electrons.Electron-phonon coupling must be strong for the Copper pair to remain intact and itmust be greater than electron-electron repulsion. The binding energy of Cooper pairsis usually weak and this is why TC is usually low.

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Inorganic Materials